10) ( r^2 - 5r - 36 )
11) ( x^2 + 5x - 50 )
12) ( 2a^2 - 14a + 24 )
13) ( 5x^2 + 50x + 80 )
solve each equation by factoring.
14) ( 4p^2 - 36p + 72 = 0 )
15) ( 6x^2 + 60x + 144 = 0 )
16) ( k^2 - 9k + 20 = 0 )
solve each equation with the quadratic formula.
17) ( 3n^2 - 4n - 84 = 0 )
18) ( 3k^2 - 11k - 4 = 0 )

Question

Accepted Answer

### Problem 10: Factor $ r^2 - 5r - 36 $
# Explanation:
## Step 1: Find two numbers
Find two numbers that multiply to $-36$ and add to $-5$. The numbers are $-9$ and $4$ (since $-9\times4 = -36$ and $-9 + 4 = -5$).
## Step 2: Factor the quadratic
Rewrite the middle term using these numbers: $ r^2 - 9r + 4r - 36 $. Then factor by grouping: $ r(r - 9) + 4(r - 9) = (r - 9)(r + 4) $.
# Answer:
$ (r - 9)(r + 4) $

### Problem 11: Factor $ x^2 + 5x - 50 $
# Explanation:
## Step 1: Find two numbers
Find two numbers that multiply to $-50$ and add to $5$. The numbers are $10$ and $-5$ (since $10\times(-5) = -50$ and $10 + (-5) = 5$).
## Step 2: Factor the quadratic
Rewrite the middle term: $ x^2 + 10x - 5x - 50 $. Factor by grouping: $ x(x + 10) - 5(x + 10) = (x + 10)(x - 5) $.
# Answer:
$ (x + 10)(x - 5) $

### Problem 12: Factor $ 2a^2 - 14a + 24 $
# Explanation:
## Step 1: Factor out the GCF
The greatest common factor (GCF) of $2a^2$, $-14a$, and $24$ is $2$. Factor out $2$: $ 2(a^2 - 7a + 12) $.
## Step 2: Factor the quadratic
Find two numbers that multiply to $12$ and add to $-7$. The numbers are $-3$ and $-4$. So, $ a^2 - 7a + 12 = (a - 3)(a - 4) $. Thus, the factored form is $ 2(a - 3)(a - 4) $.
# Answer:
$ 2(a - 3)(a - 4) $

### Problem 13: Factor $ 5x^2 + 50x + 80 $
# Explanation:
## Step 1: Factor out the GCF
The GCF of $5x^2$, $50x$, and $80$ is $5$. Factor out $5$: $ 5(x^2 + 10x + 16) $.
## Step 2: Factor the quadratic
Find two numbers that multiply to $16$ and add to $10$. The numbers are $2$ and $8$. So, $ x^2 + 10x + 16 = (x + 2)(x + 8) $. Thus, the factored form is $ 5(x + 2)(x + 8) $.
# Answer:
$ 5(x + 2)(x + 8) $

### Problem 14: Solve $ 4p^2 - 36p + 72 = 0 $ by factoring
# Explanation:
## Step 1: Factor out the GCF
The GCF of $4p^2$, $-36p$, and $72$ is $4$. Factor out $4$: $ 4(p^2 - 9p + 18) = 0 $.
## Step 2: Factor the quadratic
Find two numbers that multiply to $18$ and add to $-9$. The numbers are $-3$ and $-6$. So, $ p^2 - 9p + 18 = (p - 3)(p - 6) $. The equation becomes $ 4(p - 3)(p - 6) = 0 $.
## Step 3: Solve for $p$
Set each factor equal to zero: $ p - 3 = 0 $ or $ p - 6 = 0 $. Thus, $ p = 3 $ or $ p = 6 $.
# Answer:
$ p = 3 $ or $ p = 6 $

### Problem 15: Solve $ 6x^2 + 60x + 144 = 0 $ by factoring
# Explanation:
## Step 1: Factor out the GCF
The GCF of $6x^2$, $60x$, and $144$ is $6$. Factor out $6$: $ 6(x^2 + 10x + 24) = 0 $.
## Step 2: Factor the quadratic
Find two numbers that multiply to $24$ and add to $10$. The numbers are $4$ and $6$. So, $ x^2 + 10x + 24 = (x + 4)(x + 6) $. The equation becomes $ 6(x + 4)(x + 6) = 0 $.
## Step 3: Solve for $x$
Set each factor equal to zero: $ x + 4 = 0 $ or $ x + 6 = 0 $. Thus, $ x = -4 $ or $ x = -6 $.
# Answer:
$ x = -4 $ or $ x = -6 $

### Problem 16: Solve $ k^2 - 9k + 20 = 0 $ by factoring
# Explanation:
## Step 1: Factor the quadratic
Find two numbers that multiply to $20$ and add to $-9$. The numbers are $-4$ and $-5$. So, $ k^2 - 9k + 20 = (k - 4)(k - 5) = 0 $.
## Step 2: Solve for $k$
Set each factor equal to zero: $ k - 4 = 0 $ or $ k - 5 = 0 $. Thus, $ k = 4 $ or $ k = 5 $.
# Answer:
$ k = 4 $ or $ k = 5 $

### Problem 17: Solve $ 3n^2 - 4n - 84 = 0 $ using the quadratic formula
The quadratic formula is $ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ for an equation $ ax^2 + bx + c = 0 $.
# Explanation:
## Step 1: Identify $a$, $b$, and $c$
For $ 3n^2 - 4n - 84 = 0 $, $ a = 3 $, $ b = -4 $, $ c = -84 $.
## Step 2: Calculate the discriminant
The discriminant $ D = b^2 - 4ac = (-4)^2 - 4(3)(-84) = 16 + 1008 = 1024 $.
## Step 3: Apply the quadratic formula
$ n = \frac{-(-4) \pm \sqrt{1024}}{2(3)} = \frac{4 \pm 32}{6} $.
## Step 4: Find the solutions
For the plus sign: $ n = \frac{4 + 32}{6} = \frac{36}{6} = 6 $. For the minus sign: $ n = \frac{4 - 32}{6} = \frac{-28}{6} = -\frac{14}{3} $.
# Answer:
$ n = 6 $ or $ n = -\frac{14}{3} $

### Problem 18: Solve $ 3k^2 - 11k - 4 = 0 $ using the quadratic formula
The quadratic formula is $ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ for an equation $ ax^2 + bx + c = 0 $.
# Explanation:
## Step 1: Identify $a$, $b$, and $c$
For $ 3k^2 - 11k - 4 = 0 $, $ a = 3 $, $ b = -11 $, $ c = -4 $.
## Step 2: Calculate the discriminant
The discriminant $ D = b^2 - 4ac = (-11)^2 - 4(3)(-4) = 121 + 48 = 169 $.
## Step 3: Apply the quadratic formula
$ k = \frac{-(-11) \pm \sqrt{169}}{2(3)} = \frac{11 \pm 13}{6} $.
## Step 4: Find the solutions
For the plus sign: $ k = \frac{11 + 13}{6} = \frac{24}{6} = 4 $. For the minus sign: $ k = \frac{11 - 13}{6} = \frac{-2}{6} = -\frac{1}{3} $.
# Answer:
$ k = 4 $ or $ k = -\frac{1}{3} $

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QUESTION IMAGE

Question

Explanation:

Problem 10: Factor \( r^2 - 5r - 36 \)

Step 1: Find two numbers

Step 2: Factor the quadratic

Step 1: Find two numbers

Step 2: Factor the quadratic

Step 1: Factor out the GCF

Step 2: Factor the quadratic

Answer:

Problem 11: Factor \( x^2 + 5x - 50 \)